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### Problem 1.37: Writing the dual of each equation

**Duality Principle:** 
In set theory, the duality principle states that if we interchange ∪ (union) with ∩ (intersection) and U (universal set) with ∅ (empty set) in a given identity, we obtain another valid identity called the dual.

**(a) Original equation:**
\[
A = (B^C \cap A) \cup (A \cap B)
\]
**Dual:**
Applying the duality principle:
- ∪ becomes ∩
- ∩ becomes ∪
- \(B^C\) (complement of B) stays the same.

The dual of this equation is:
\[
A = (B^C \cup A) \cap (A \cup B)
\]

**(b) Original equation:**
\[
(A \cap B) \cup (A^C \cap B) \cup (A \cap B^C) \cup (A^C \cap B^C) = U
\]
**Dual:**
- ∪ becomes ∩
- ∩ becomes ∪
- U (universal set) becomes ∅ (empty set).

The dual of this equation is:
\[
(A \cup B) \cap (A^C \cup B) \cap (A \cup B^C) \cap (A^C \cup B^C) = \emptyset
\]

---

### Problem 1.38: Proving the set identities using set laws

**(a) Prove \( (A \cap B) \cup (A \cap B^C) = A \)**

**Proof:**
1. Distribute \(A\):
   \[
   (A \cap B) \cup (A \cap B^C) = A \cap (B \cup B^C)
   \]
2. By the Complement Law, \( B \cup B^C = U \) (the universal set).
   \[
   A \cap U = A
   \]
Thus, the equation is proven:
\[
(A \cap B) \cup (A \cap B^C) = A
\]

---

**(b) Prove \( A \cup B = (A \cap B^C) \cup (A^C \cap B) \cup (A \cap B) \)**

**Proof:**
1. Note that the right-hand side is the union of three disjoint sets: \( (A \cap B^C), (A^C \cap B), (A \cap B) \).
   - \( (A \cap B^C) \) is the part of \( A \) that does not intersect \( B \).
   - \( (A^C \cap B) \) is the part of \( B \) that does not intersect \( A \).
   - \( (A \cap B) \) is the intersection of \( A \) and \( B \).
2. Together, these three regions form the union \( A \cup B \), covering all elements in either \( A \), \( B \), or both.

Thus, the equation is proven:
\[
A \cup B = (A \cap B^C) \cup (A^C \cap B) \cup (A \cap B)
\]

---

Would you like further details or clarifications? Here are some related questions for additional practice:

1. What is the dual of \( A \cup (A^C \cap B) = B \)?
2. Can you write the dual of \( A \cup A^C = U \)?
3. How would you apply De Morgan's Laws to prove dual identities?
4. What is the result of the dual of \( A \cap A^C = \emptyset \)?
5. Can you prove \( (A \cup B)^C = A^C \cap B^C \) using De Morgan's Law?

**Tip:** Always remember to use the basic set laws, such as Distributive, Complement, and Identity laws, when proving identities in set theory!

Write the dual of each equation and prove the set identities using the laws from Table 1-1.

Learn how to apply the duality principle in set theory and prove set identities using key laws such as the Complement and Distributive Laws. This example walks through the process with step-by-step solutions.

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